In this paper we consider a Hamiltonian H on P_2(R^{2d} ), the set of probability measures with finite quadratic moments on the phase space R^{2d} = R^d × R^d , which is a metric space when endowed with the Wasserstein distance W_2. We study the initial value problem dμ_t/dt+∇·(J_dv_tμ_t ) = 0, where J_d is the canonical symplectic matrix, μ_0 is prescribed, and v_t is a tangent vector to P_2(R^{2d}) at μ_t , belonging to ∂H(μ_t ), the subdifferential of H at μ_t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ_0 is absolutely continuous. It ensures that μ_t remains absolutely continuous and v_t = ∇H(μ_t ) is the element of minimal norm in ∂H(μt ). The second method handles any initial measure μ_0. If we further assume that H is λ-convex, proper, and lowersemicontinuous on P_2(R^{2d} ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, namely H(μ_t ) = H(μ_0).

Hamiltonian ODEs in the Wasserstein space of probability measures

AMBROSIO, Luigi;
2008-01-01

Abstract

In this paper we consider a Hamiltonian H on P_2(R^{2d} ), the set of probability measures with finite quadratic moments on the phase space R^{2d} = R^d × R^d , which is a metric space when endowed with the Wasserstein distance W_2. We study the initial value problem dμ_t/dt+∇·(J_dv_tμ_t ) = 0, where J_d is the canonical symplectic matrix, μ_0 is prescribed, and v_t is a tangent vector to P_2(R^{2d}) at μ_t , belonging to ∂H(μ_t ), the subdifferential of H at μ_t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ_0 is absolutely continuous. It ensures that μ_t remains absolutely continuous and v_t = ∇H(μ_t ) is the element of minimal norm in ∂H(μt ). The second method handles any initial measure μ_0. If we further assume that H is λ-convex, proper, and lowersemicontinuous on P_2(R^{2d} ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, namely H(μ_t ) = H(μ_0).
File in questo prodotto:
File Dimensione Formato  
Ambrosio_CPAM.pdf

Accesso chiuso

Tipologia: Altro materiale allegato
Licenza: Non pubblico
Dimensione 298.69 kB
Formato Adobe PDF
298.69 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/502
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 81
  • ???jsp.display-item.citation.isi??? 71
social impact